Learing & Lovin' Math - Locker Room Problem

Today was the day we investigated the famous locker room problem in CMP:

Imagine in a school, there are 1000 lockers.  On the first day of school, all the kids line up. Kid 1 runs through and opens all the lockers.  Kid 2 runs through next and closes Lockers 2,4,6,8 and so on.  Kid 3 runs through and changes the state of (closes the open ones, opens the closed ones) of lockers 3,6,9,12 etc.

In the end, after the 1000th kid has run through, which lockers will be open? Why?

Warning: Answer is given below. DO NOT SCROLL if you wish to first solve the problem yourself......

This problem has historically been the bugbear, to the point, we have a Locker Room Problem Hall of Fame at Al Qamar.  

Apparently three kids in class had already "read" up their math books and solved it two weeks in advance - my math nerds. One was walking up and down the corridors lost in his own thoughts, opening and closing locker doors in his head until he came up with the answer.

Today, the rest of them  were bent over their books coming up with various diagrams and ideas to solve the problem. 

They were initially overwhelmed by the fact that they had to go through the motion with a 1000 students and 1000 doors!

The hint by G.Polya to solve a simplified version of the problem and search for patterns helped them. They chose to work with 10-30 doors to seek patterns.

... And then they were buzzing with multiples - open, close, open..

"Hey the prime numbers will be closed".. came the first observation. Number 1 opens them and they close themselves. They have only two factors.. an Aha! moment there.

Soon they solved the problem for 10-30 doors, checked the open doors and found 1,4,9,16,25 to be open. "Square numbers", came the answer from different teams.  

We further explored why square number doors were open - "because they have an odd number of factors, they don't have all factors pairs. So the open door will remain open " came the replies.

One child had a different way of approaching the problem.

She simply wrote down numbers 1-100. Made some sort of long calculations, some scribbles, ended up tearing the paper due to excessive erasing, got another set of numbers, stared at them and then, looking up, told me "The answer is square numbers". On asking her the reason, she replied "It's because there are no two two people ,no three three people and so on." Now scratch your heads to figure that out.

I had to ask her to explain atleast three times before I got it.

"Look Aunty, 3*2=6, there is a person 3 and 2 but 2*2=4, 3*3=9, there are no 2 of the same people, so only one will touch the door. So after the other factors close it that single person will open it"

She worked out the idea and got her answer  the other way round than done by most kids, the teacher included.

This is how Maths should be learned!

- R. Riaz